As in [1]}, we use an AR(1) process [2]} to model the correlation between \(x^{(l)}_n\)  and \(x^{(l-1)}_n\) , i.e.,
\(\begin{aligned}x^{(l)}_n&=\alpha x^{(l-1)}_n + \sqrt{1-\alpha ^2} \vartheta ^{(l)}_n \\p(x^{(l)}_n|x^{(l-1)}_n)&=\mathcal {N} (x^{(l)}_n|\alpha x^{(l-1)}_n,(1-\alpha ^2) {{\gamma }^{-1}_n}), l >1 \\p(x^{(1)}_n)&=\mathcal {N} (x^{(1)}_n|0, {{\gamma }^{-1}_n}),\end{aligned}\) 
